An underdamped and delayed tri-stable model-based stochastic resonance

Stochastic resonance(SR) is investigated in an underdamped tri-stable potential system driven by Gaussian colored noise and a periodic excitation, where both displacement and velocity time-delayed states feedback are considered. It is challenging to study SR in a second-order delayed multi-stable system analytically. In this paper, the improved energy envelope stochastic average method is developed to derive the analytical expressions of stationary probability density(SPD)and spectral amplification. The effects of noise intensity, damping coefficient, and time delay on SR are analyzed. The results show that the shapes of joint SPD can be adjusted to the desired structure by choosing the time delay and feedback gains. For fixed time delay, the SR peak is increased for negative displacement or velocity feedback gain. Meanwhile, the SR peak is decreased while the optimal noise intensity increases with increasing correlation time of noise. The Monte Carlo simulations(MCS) confirm the effectiveness of the theoretical results.

Logical stochastic resonance in a cross-bifurcation non-smooth system

This paper investigates logical stochastic resonance(LSR)in a cross-bifurcation non-smooth system driven by Gaussian colored noise.In this system,a bifurcation parameter triggers a transition between monostability,bistability and tristability.By using Novikov's theorem and the unified colored noise approximation method,the approximate Fokker-Planck equation is obtained.Then we derive the generalized potential function and the transition rates to analyze the LSR phenomenon using numerical simulations.We simulate the logic operation of the system in the bistable and tristable regions respectively.We assess the impact of Gaussian colored noise on the LSR and discover that the reliability of the logic response depends on the noise strength and the bifurcation parameter.Furthermore,it is found that the bistable region has a more extensive parameter range to produce reliable logic operation compared with the tristable region,since the tristable region is more sensitive to noise than the bistable one.

Research and application of composite stochastic resonance in enhancement detection

Aiming at the problem that the intermediate potential part of the traditional bistable stochastic resonance model cannot be adjusted independently, a new composite stochastic resonance(NCSR) model is proposed by combining the Woods–Saxon(WS) model and the improved piecewise bistable model. The model retains the characteristics of the independent parameters of WS model and the improved piecewise model has no output saturation, all the parameters in the new model have no coupling characteristics. Under α stable noise environment, the new model is used to detect periodic signal and aperiodic signal, the detection results indicate that the new model has higher noise utilization and better detection effect.Finally, the new model is applied to image denoising, the results showed that under the same conditions, the output peak signal-to-noise ratio(PSNR) and the correlation number of NCSR method is higher than that of other commonly used linear denoising methods and improved piecewise SR methods, the effectiveness of the new model is verified.

Adaptive Bistable Stochastic Resonance Based Weak Signal Reception in Additive Laplacian Noise

Weak signal reception is a very important and challenging problem for communication systems especially in the presence of non-Gaussian noise,and in which case the performance of optimal linear correlated receiver degrades dramatically.Aiming at this,a novel uncorrelated reception scheme based on adaptive bistable stochastic resonance(ABSR)for a weak signal in additive Laplacian noise is investigated.By analyzing the key issue that the quantitative cooperative resonance matching relationship between the characteristics of the noisy signal and the nonlinear bistable system,an analytical expression of the bistable system parameters is derived.On this basis,by means of bistable system parameters self-adaptive adjustment,the counterintuitive stochastic resonance(SR)phenomenon can be easily generated at which the random noise is changed into a benefit to assist signal transmission.Finally,it is demonstrated that approximately 8dB bit error ratio(BER)performance improvement for the ABSR-based uncorrelated receiver when compared with the traditional uncorrelated receiver at low signal to noise ratio(SNR)conditions varying from-30dB to-5dB.

Weak signal detection method based on novel composite multistable stochastic resonance

The weak signal detection method based on stochastic resonance is usually used to extract and identify the weak characteristic signal submerged in strong noise by using the noise energy transfer mechanism.We propose a novel composite multistable stochastic-resonance(NCMSR)model combining the Gaussian potential model and an improved bistable model.Compared with the traditional multistable stochastic resonance method,all the parameters in the novel model have no symmetry,the output signal-to-noise ratio can be optimized and the output amplitude can be improved by adjusting the system parameters.The model retains the advantages of continuity and constraint of the Gaussian potential model and the advantages of the improved bistable model without output saturation,the NCMSR model has a higher utilization of noise.Taking the output signal-to-noise ratio as the index,weak periodic signal is detected based on the NCMSR model in Gaussian noise andαnoise environment respectively,and the detection effect is good.The application of NCMSR to the actual detection of bearing fault signals can realize the fault detection of bearing inner race and outer race.The outstanding advantages of this method in weak signal detection are verified,which provides a theoretical basis for industrial practical applications.

Inverse stochastic resonance in modular neural network with synaptic plasticity

This work explores the inverse stochastic resonance(ISR) induced by bounded noise and the multiple inverse stochastic resonance induced by time delay by constructing a modular neural network, where the modified Oja’s synaptic learning rule is employed to characterize synaptic plasticity in this network. Meanwhile, the effects of synaptic plasticity on the ISR dynamics are investigated. Through numerical simulations, it is found that the mean firing rate curve under the influence of bounded noise has an inverted bell-like shape, which implies the appearance of ISR. Moreover, synaptic plasticity with smaller learning rate strengthens this ISR phenomenon, while synaptic plasticity with larger learning rate weakens or even destroys it. On the other hand, the mean firing rate curve under the influence of time delay is found to exhibit a decaying oscillatory process, which represents the emergence of multiple ISR. However, the multiple ISR phenomenon gradually weakens until it disappears with increasing noise amplitude. On the same time, synaptic plasticity with smaller learning rate also weakens this multiple ISR phenomenon, while synaptic plasticity with larger learning rate strengthens it. Furthermore, we find that changes of synaptic learning rate can induce the emergence of ISR phenomenon. We hope these obtained results would provide new insights into the study of ISR in neuroscience.

Analytical and NumericalMethods to Study the MFPT and SR of a Stochastic Tumor-Immune Model

The Mean First-Passage Time (MFPT) and Stochastic Resonance (SR) of a stochastic tumor-immune model withnoise perturbation are discussed in this paper. Firstly, considering environmental perturbation, Gaussian whitenoise and Gaussian colored noise are introduced into a tumor growth model under immune surveillance. Asfollows, the long-time evolution of the tumor characterized by the Stationary Probability Density (SPD) and MFPTis obtained in theory on the basis of the Approximated Fokker-Planck Equation (AFPE). Herein the recurrenceof the tumor from the extinction state to the tumor-present state is more concerned in this paper. A moreefficient algorithmof Back-Propagation Neural Network (BPNN) is utilized in order to testify the correction of thetheoretical SPDandMFPT.With the existence of aweak signal, the functional relationship between Signal-to-NoiseRatio (SNR), noise intensities and correlation time is also studied. Numerical results show that both multiplicativeGaussian colored noise and additive Gaussian white noise can promote the extinction of the tumors, and themultiplicative Gaussian colored noise can lead to the resonance-like peak on MFPT curves, while the increasingintensity of the additiveGaussian white noise results in theminimum of MFPT. In addition, the correlation timesare negatively correlated with MFPT. As for the SNR, we find the intensities of both the Gaussian white noise andthe Gaussian colored noise, as well as their correlation intensity can induce SR. Especially, SNR is monotonouslyincreased in the case ofGaussian white noisewith the change of the correlation time.At last, the optimal parametersin BPNN structure are analyzed for MFPT from three aspects: the penalty factors, the number of neural networklayers and the number of nodes in each layer.

Stochastic resonance in a bias linear system with multiplicative and additive noise

In this paper, the stochastic resonance in a bias linear system subjected multiplicative and additive dichotomous noise is investigated. Using the linear-response theory and the properties of the dichotomous noise, this paper finds the exact expressions for the first two moments and the signal-to-noise ratio （SNR）. It is shown that the SNR is a non-monotonic function of the correlation time of the multiplicative and additive noise, and it varies non-monotonously with the intensity and asymmetry of the multiplicative noise as well as the external field frequency. Moreover, the SNR depends on the system bias, the intensity of the cross noise between the multiplicative and additive noise, and the strength and asymmetry of the additive noise.

Enhanced Detection of Rolling Element Bearing Fault Based on Stochastic Resonance

Early bearing faults can generate a series of weak impacts. All the influence factors in measurement may degrade the vibration signal. Currently, bearing fault enhanced detection method based on stochastic resonance（SR） is implemented by expensive computation and demands high sampling rate, which requires high quality software and hardware for fault diagnosis. In order to extract bearing characteristic frequencies component, SR normalized scale transform procedures are presented and a circuit module is designed based on parameter-tuning bistable SR. In the simulation test, discrete and analog sinusoidal signals under heavy noise are enhanced by SR normalized scale transform and circuit module respectively. Two bearing fault enhanced detection strategies are proposed. One is realized by pure computation with normalized scale transform for sampled vibration signal, and the other is carried out by designed SR hardware with circuit module for analog vibration signal directly. The first strategy is flexible for discrete signal processing, and the second strategy demands much lower sampling frequency and less computational cost. The application results of the two strategies on bearing inner race fault detection of a test rig show that the local signal to noise ratio of the characteristic components obtained by the proposed methods are enhanced by about 50% compared with the band pass envelope analysis for the bearing with weaker fault. In addition, helicopter transmission bearing fault detection validates the effectiveness of the enhanced detection strategy with hardware. The combination of SR normalized scale transform and circuit module can meet the need of different application fields or conditions, thus providing a practical scheme for enhanced detection of bearing fault.

Application of Stochastic Resonance Signal Recovery

Stochastic resonance（SR） enhances the nonlinear system behavior with the assistance of noise, including the sensitivity and selectivity of the response to the exterior stimulus. The energy-transfer mechanism makes the weak information revealed in the output spectrum, while the time-waveform is distorted. The distortion analysis was made both from the particle＇s dynan-fics and signal processing. The factors causing the deviation in the output are presented and the function of the recovery system is proposed. By the investigation of the particle＇s motion track in the bistable system and the suggested recovery system, the influences of noise and system parameters on the recovery course were discussed. Moreover, the pulse distortion appearing the recovery waveform caused by the particle＇s transitions at the bistable potential＇ inflexions was explained. Due to different characteristics, cascaded-bistable SR or mono-stable SR was introduced to process different types of signals. The final recovery signal is just the suggested recovery system＇s response to the SR output. Meanwhile, the recovery system is optional, as parameter-tuned or parameter-fixed. Since the method requires no average processing, it is applicable to a single sample with limited length. The numerical simulations reveal that the SR recovery method can recover the waveform containing weak information submerged ha noise effectively. The engineering application to the vibration analysis of metal cutting chose the combination of mono-stable SR and the parameter-fixed recovery system. Because the optimal SR state is not required strongly, the system parameters are tuned in a wider range than the traditional SR processing methods.

Parameter-induced stochastic resonance with a periodic signal

In this paper conventional stochastic resonance （CSR） is realized by adding the noise intensity. This demonstrates that tuning the system parameters with fixed noise can make the noise play a constructive role and realize parameter- induced stochastic resonance （PSR）. PSR can be interpreted as changing the intrinsic characteristic of the dynamical system to yield the cooperative effect between the stochastic-subjected nonlinear system and the external periodic force. This can be realized at any noise intensity, which greatly differs from CSR that is realized under the condition of the initial noise intensity not greater than the resonance level. Moreover, it is proved that PSR is different from the optimization of system parameters.

Stochastic resonance in a monostable system driven by square-wave signal and dichotomous noise

This paper investigates the stochastic resonance in a monostable system driven by square-wave signal, asymmetric dichotomous noise as well as by multiplicative and additive white noise. By the use of the properties of the dichotomous noise, it obtains the expressions of the signal-to-noise ratio under the adiabatic approximation condition. It finds that the signal-to-noise ratio is a non-monotonic function of the asymmetry of the dichotomous noise, and which varies non- monotonously with the intensity of the multiplicative and additive noise as well as the system parameters. Moreover, the signal-to-noise ratio depends on the correlation rate and intensity of the dichotomous noise.

Stochastic resonance in a time-delayed bistable system subject to multiplicative and additive noise

This paper investigates the stochastic resonance in a time-delayed bistable system subjected to multiplicative and additive white noise and asymmetric dichotomous noise. Under the adiabatic approximation condition, the expression of the signal-to-noise ratio （SNR） is obtained. It finds that the SNR is a non-monotonic function of the delayed times, of the amplitude of the driving square-wave signal, as well as of the asymmetry of the dichotomous noise. In addition, the SNR varies non-monotonously with the intensities of the multiplicative and additive noise as well as the system parameters. Moreover, the SNR depends non-monotonically on the correlate rate of the dichotomous noise.

Stochastic resonance in a time-delayed mono-stable system with correlated multiplicative and additive white noise

This paper considers the stochastic resonance for a time-delayed mono-stable system, driven by correlated multiplicative and additive white noise. It finds that the output signal-to-noise ratio （SNR） varies non-monotonically with the delayed times. The SNR varies non-monotonically with the increase of the intensities of the multiplicative and additive noise, with the increase of the correlation strength between the two noises, as well as with the system parameter.

Stochastic resonance in a stochastic bistable system with additive noises and square-wave signal

This paper considers the stochastic resonance in a stochastic bistable system driven by a periodic square-wave signal and a static force as well as by additive white noise and dichotomous noise from the viewpoint of signal-to-noise ratio. It finds that the signal-to-noise ratio appears as stochastic resonance behaviour when it is plotted as a function of the noise strength of the white noise and dichotomous noise, as a function of the system parameters, or as a function of the static force. Moreover, the influence of the strength of the stochastic potential force and the correlation rate of the dichotomous noise on the signal-to-noise ratio is investigated.

Delay-aided stochastic multiresonances on scale-free FitzHugh-Nagumo neuronal networks

The stochastic resonance in paced time-delayed scale-free FitzHugh--Nagumo （FHN） neuronal networks is investigated. We show that an intermediate intensity of additive noise is able to optimally assist the pacemaker in imposing its rhythm on the whole ensemble. Furthermore, we reveal that appropriately tuned delays can induce stochastic multiresonances, appearing at every integer multiple of the pacemaker＇s oscillation period. We conclude that fine-tuned delay lengths and locally acting pacemakers are vital for ensuring optimal conditions for stochastic resonance on complex neuronal networks.

Weak thruster fault detection for AUV based on stochastic resonance and wavelet reconstruction

When the bi-stable stochastic resonance method was applied to enhance weak thruster fault for autonomous underwater vehicle(AUV), the enhancement performance could not satisfy the detection requirement of weak thruster fault. As for this problem, a fault feature enhancement method based on mono-stable stochastic resonance was proposed. In the method, in order to improve the enhancement performance of weak thruster fault feature, the conventional bi-stable potential function was changed to mono-stable potential function which was more suitable for aperiodic signals. Furthermore, when particle swarm optimization was adopted to adjust the parameters of mono-stable stochastic resonance system, the global convergent time would be long. An improved particle swarm optimization method was developed by changing the linear inertial weighted function as nonlinear function with cosine function, so as to reduce the global convergent time. In addition, when the conventional wavelet reconstruction method was adopted to detect the weak thruster fault, undetected fault or false alarm may occur. In order to successfully detect the weak thruster fault, a weak thruster detection method was proposed based on the integration of stochastic resonance and wavelet reconstruction. In the method, the optimal reconstruction scale was determined by comparing wavelet entropies corresponding to each decomposition scale. Finally, pool-experiments were performed on AUV with thruster fault. The effectiveness of the proposed mono-stable stochastic resonance method in enhancing fault feature and reducing the global convergent time was demonstrated in comparison with particle swarm optimization based bi-stochastic resonance method. Furthermore, the effectiveness of the proposed fault detection method was illustrated in comparison with the conventional wavelet reconstruction.

Stochastic resonance in coupled weakly-damped bistable oscillators subjected to additive and multiplicative noises

With coupled weakly-damped periodically driven bistable oscillators subjected to additive and multiplicative noises under concern, the objective of this paper is to check to what extent the resonant point predicted by the Gaussian distribution assumption can approximate the simulated one. The investigation based on the dynamical mean-field approx- imation and the direct simulation demonstrates that the pre- dicted resonant point and the simulated one are basically co- incident for the case of pure additive noise, but for the case including multiplicative noise the situation becomes some- what complex. Specifically speaking, when stochastic res- onance （SR） is observed by changing the additive noise in- tensity, the predicted resonant point is lower than the sim- ulated one; nevertheless, when SR is observed by chang- ing the multiplicative noise intensity, the predicted resonant point is higher than the simulated one. Our observations im- ply that the Gaussian distribution assumption can not exactly describe the actual situation, but it is useful to some extent in predicting the low-frequency stochastic resonance of the coupled weakly-damped bistable oscillator.

Stochastic resonance in an under-damped bistable system driven by harmonic mixing signal

Stochastic resonance（SR） is studied in an under-damped bistable system driven by the harmonic mixing signal and Gaussian white noise. Using the linear response theory（LRT）, the expressions of the spectral amplification at fundamental and higher-order harmonic are obtained. The effects of damping coefficient, noise intensity, signal amplitude, and frequency on spectral amplifications are explored. Meanwhile, the power spectral density（PSD） and signal-to-noise ratio（SNR） are calculated to quantify SR and verify the theoretical results. The SNRs at the first and second harmonics exhibit a minimum first and a maximum later with increasing noise intensity. That is, both of the noise-induced suppression and resonance can be observed by choosing proper system parameters. Especially, when the ratio of the second harmonic amplitude to the fundamental one takes a large value, the SNR at the fundamental harmonic is a monotonic function of noise intensity and the SR phenomenon disappears.

Stochastic resonance in a single-mode laser driven by quadratic pump noise and amplitude-modulated signal

This paper investigates the phenomenon of stochastic resonance in a single-mode laser driven by quadratic pump noise and amplitude-modulated signal. A new linear approximation approach is advanced to calculate the signal-to-noise ratio. In the linear approximation only the drift term is linearized, the multiplicative noise term is unchangeable. It is found that there appears not only the standard form of stochastic resonance but also the broad sense of stochastic resonance, especially stochastic multiresonance appears in the curve of signal-to-noise ratio as a function of coupling strength A between the real and imaginary parts of the pump noise.